Group #21436732 — subgroups

Subgroups — #21436732

Order 7552.

426 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838074	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837749	C₂	1	3	yes	maximal, complemented
4	#16837683	C₂	1	3	yes	maximal, complemented
5	#16837598	C₂	1	3	yes	maximal, complemented
6	C₄×D₄₇₂	C₂	1	3	yes	maximal, complemented
7	C₈×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745689	—	2	1	no	—
9	C₂×Q₈×D₅₉	—	2	1	no	—
10	C₂×D₄₇₂	C₂²	1	3	yes	—
11	#14273727	C₂²	1	3	yes	—
12	#14273724	C₂²	1	3	yes	—
13	#14273707	—	2	3	no	—
14	#14273702	—	2	3	no	—
15	C₄×D₂₃₆	C₂²	1	3	yes	—
16	C₄×D₂₃₆	C₂²	1	3	yes	—
17	#14273661	C₂²	1	3	yes	—
18	#14273649	C₂²	1	6	yes	—
19	#14273649	C₂²	1	1	yes	characteristic
20	#14273649	C₂²	1	3	yes	—
21	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₉	C₂²	1	3	yes	—
23	C₄×C₈×C₅₉	C₂²	1	3	yes	—
24	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
25	#11745689	—	4	4	no	—
26	Q₈×D₅₉	—	4	4	no	—
27	C₂×D₂₃₆	—	2	3	no	—
28	C₂×D₂₃₆	—	2	3	no	—
29	C₂×D₂₃₆	C₂³	1	3	yes	—
30	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
31	C₂×C₄×D₅₉	—	2	3	no	—
32	C₂×C₄×D₅₉	—	2	3	no	—
33	C₂×Dic₁₁₈	—	2	3	no	—
34	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₅₉	C₂³	1	3	yes	—
36	C₄⋊C₄×C₅₉	D₄	1	3	yes	—
37	C₄²×C₅₉	C₂³	1	3	yes	—
38	(C₂×D₅₉)⋊C₄	—	2	3	no	—
39	(C₂×D₅₉)⋊C₄	—	2	3	no	—
40	C₄⋊Dic₅₉	—	2	3	no	—
41	C₄⋊Dic₅₉	C₂³	1	3	yes	—
42	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
43	D₄₇₂	—	2	6	no	—
44	C₂²×D₅₉	—	2	1	no	—
45	C₂²×D₅₉	—	2	3	no	—
46	Q₈×C₅₉	D₈	1	4	yes	complemented
47	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
48	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₅₉	—	2	3	no	—
51	C₂×Dic₅₉	—	2	1	no	—
52	D₂₃₆	—	2	3	no	—
53	D₂₃₆	—	2	1	no	—
54	D₂₃₆	—	2	3	no	—
55	D₂₃₆	—	4	6	no	—
56	D₂₃₆	—	8	3	no	—
57	D₂₃₆	#41	1	2	yes	complemented
58	C₄×D₅₉	—	4	6	no	—
59	C₄×D₅₉	—	4	6	no	—
60	Dic₁₁₈	—	4	6	no	—
61	C₈×C₅₉	—	2	3	no	—
62	C₈×C₅₉	—	2	1	no	—
63	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
64	C₂×D₅₉	—	2	2	no	—
65	C₂×D₅₉	—	4	1	no	—
66	C₂×D₅₉	—	4	1	no	—
67	C₂×D₅₉	—	4	3	no	—
68	C₂×D₅₉	—	4	3	no	—
69	C₂×D₅₉	—	4	3	no	—
70	C₄×C₅₉	—	2	3	no	—
71	C₄×C₅₉	C₂×D₈	1	6	yes	—
72	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
73	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
74	Dic₅₉	—	4	3	no	—
75	Dic₅₉	—	4	1	no	—
76	#3358	—	59	1	no	maximal, Sylow
77	C₂×C₅₉	#548	1	1	yes	characteristic
78	C₂×C₅₉	C₂²×D₈	1	1	yes	characteristic
79	C₂×C₅₉	#577	1	1	yes	characteristic
80	D₅₉	—	4	2	no	—
81	D₅₉	—	8	3	no	—
82	#550	—	59	1	no	—
83	D₄×Q₈	—	59	1	no	—
84	#486	—	59	3	no	—
85	#474	—	59	3	no	—
86	#459	—	59	3	no	—
87	C₈×Q₈	D₅₉	1	1	yes	characteristic, complemented
88	C₄×D₈	—	59	3	no	—
89	C₅₉	#3358	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	118	1	no	—
91	C₂²×Q₈	—	118	1	no	—
92	C₂×D₈	—	59	3	no	—
93	C₄⋊Q₈	—	59	3	no	—
94	#127	—	59	3	no	—
95	C₂²⋊Q₈	—	118	3	no	—
96	#121	—	118	3	no	—
97	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
98	C₄×D₄	—	59	3	no	—
99	C₄×D₄	—	59	3	no	—
100	#107	—	59	3	no	—
101	C₄⋊C₈	C₂×D₅₉	1	3	yes	—
102	#102	—	59	1	no	—
103	#102	—	59	3	no	—
104	#102	—	59	6	no	—
105	C₄×C₈	C₂×D₅₉	1	3	yes	—
106	#41	—	236	4	no	—
107	C₂×Q₈	—	118	3	no	—
108	C₂×Q₈	—	236	4	no	—
109	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
110	C₂×D₄	—	59	1	no	—
111	C₂×D₄	—	59	3	no	—
112	C₂×D₄	—	118	3	no	—
113	C₂×D₄	—	118	3	no	—
114	C₂²×C₄	—	118	3	no	—
115	C₂²×C₄	—	118	3	no	—
116	D₈	—	118	6	no	—
117	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₅₉	1	3	yes	—
119	C₄⋊C₄	—	59	3	no	—
120	C₄⋊C₄	—	59	1	no	—
121	C₄⋊C₄	—	118	3	no	—
122	C₄⋊C₄	D₂₃₆	1	3	yes	—
123	C₂²⋊C₄	—	118	3	no	—
124	C₂²⋊C₄	—	118	3	no	—
125	C₄²	C₂²×D₅₉	1	3	yes	—
126	C₂³	—	118	1	no	—
127	C₂³	—	118	3	no	—
128	Q₈	—	236	6	no	—
129	Q₈	D₄₇₂	1	4	yes	complemented
130	D₄	—	59	2	no	—
131	D₄	—	118	3	no	—
132	D₄	—	118	1	no	—
133	D₄	—	118	3	no	—
134	D₄	—	472	3	no	—
135	D₄	—	236	6	no	—
136	C₂×C₄	—	118	1	no	—
137	C₂×C₄	—	118	3	no	—
138	C₂×C₄	—	236	6	no	—
139	C₂×C₄	—	236	6	no	—
140	C₂×C₄	C₂×D₂₃₆	1	3	yes	—
141	C₂×C₄	C₂×D₂₃₆	1	3	yes	—
142	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
143	C₈	—	2	3	no	—
144	C₈	—	2	1	no	—
145	C₂²	—	118	2	no	—
146	C₂²	—	236	3	no	—
147	C₂²	—	236	3	no	—
148	C₂²	—	236	3	no	—
149	C₂²	—	236	1	no	—
150	C₂²	—	236	1	no	—
151	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
152	C₄	—	2	3	no	—
153	C₄	—	236	1	no	—
154	C₄	—	236	3	no	—
155	C₄	C₂×D₄₇₂	1	6	yes	—
156	C₄	#41×D₅₉	1	1	yes	characteristic
157	C₄	#14273775	1	1	yes	characteristic
158	C₂	—	472	3	no	—
159	C₂	—	236	2	no	—
160	C₂	#16838064	1	1	yes	characteristic
161	C₂	C₂²×D₄₇₂	1	1	yes	characteristic
162	C₂	#16838214	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838074

C₂

yes

maximal, characteristic, complemented

Q₈×D₂₃₆

C₂

yes

maximal, characteristic, complemented

#16837749

C₂

yes

maximal, complemented

#16837683

C₂

yes

maximal, complemented

#16837598

C₂

yes

maximal, complemented

C₄×D₄₇₂

C₂

yes

maximal, complemented

C₈×Q₈×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×#11745689

—

C₂×Q₈×D₅₉

—

C₂×D₄₇₂

C₂²

yes

—

#14273727

C₂²

yes

—

#14273724

C₂²

yes

—

#14273707

—

#14273702

—

C₄×D₂₃₆

C₂²

yes

—

C₄×D₂₃₆

C₂²

yes

—

#14273661

C₂²

yes

—

#14273649

C₂²

yes

—

#14273649

C₂²

yes

characteristic

#14273649

C₂²

yes

—

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

#11745689

—

Q₈×D₅₉

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

C₂³

yes

—

C₂×D₂₃₆

C₂³

yes

characteristic

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

C₄⋊Dic₅₉

C₂³

yes

characteristic

D₄₇₂

—

C₂²×D₅₉

—

C₂²×D₅₉

—

Q₈×C₅₉

D₈

yes

complemented

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

#41

yes

complemented

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₈

yes

—

C₄×C₅₉

C₂×#41

yes

characteristic

C₄×C₅₉

2₋¹⁺⁴

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂²×D₈

yes

characteristic

C₂×C₅₉

#577

yes

characteristic

D₅₉

—

D₅₉

—

#550

—

D₄×Q₈

—

#486

—

#474

—

#459

—

C₈×Q₈

D₅₉

yes

characteristic, complemented

C₄×D₈

—

C₅₉

#3358

yes

characteristic, complemented, Sylow

C₂×#41

—

118

—

C₂²×Q₈

—

118

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂²⋊Q₈

—

118

—

#121

—

118

—

C₄×Q₈

C₂×D₅₉

yes

characteristic

C₄×D₄

—

C₄×D₄

—

100

#107

—

101

C₄⋊C₈

C₂×D₅₉

yes

—

102

#102

—

103

#102

—

104

#102

—

105

C₄×C₈

C₂×D₅₉

yes

—

106

#41

—

236

—

107

C₂×Q₈

—

118

—

108

C₂×Q₈

—

236

—

109

C₂×Q₈

D₂₃₆

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

118

—

113

C₂×D₄

—

118

—

114

C₂²×C₄

—

118

—

115

C₂²×C₄

—

118

—

116

D₈

—

118

—

117

C₂×C₈

C₂²×D₅₉

yes

characteristic

118

C₂×C₈

C₂²×D₅₉

yes

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

118

—

122

C₄⋊C₄

D₂₃₆

yes

—

123

C₂²⋊C₄

—

118

—

124

C₂²⋊C₄

—

118

—

125

C₄²

C₂²×D₅₉

yes

—

126

C₂³

—

118

—

127

C₂³

—

118

—

128

Q₈

—

236

—

129

Q₈

D₄₇₂

yes

complemented

130

D₄

—

131

D₄

—

118

—

132

D₄

—

118

—

133

D₄

—

118

—

134

D₄

—

472

—

135

D₄

—

236

—

136

C₂×C₄

—

118

—

137

C₂×C₄

—

118

—

138

C₂×C₄

—

236

—

139

C₂×C₄

—

236

—

140

C₂×C₄

C₂×D₂₃₆

yes

—

141

C₂×C₄

C₂×D₂₃₆

yes

—

142

C₂×C₄

C₂³×D₅₉

yes

characteristic

143

C₈

—

144

C₈

—

145

C₂²

—

118

—

146

C₂²

—

236

—

147

C₂²

—

236

—

148

C₂²

—

236

—

149

C₂²

—

236

—

150

C₂²

—

236

—

151

C₂²

C₂²×D₂₃₆

yes

characteristic

152

C₄

—

153

C₄

—

236

—

154

C₄

—

236

—

155

C₄

C₂×D₄₇₂

yes

—

156

C₄

#41×D₅₉

yes

characteristic

157

C₄

#14273775

yes

characteristic

158

C₂

—

472

—

159

C₂

—

236

—

160

C₂

#16838064

yes

characteristic

161

C₂

C₂²×D₄₇₂

yes

characteristic

162

C₂

#16838214

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₈×Q₈×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	C₈×Q₈	D₅₉
hypocenter	C₅₉	#3358
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₈×Q₈×C₅₉

C₂

socle

C₂²×C₅₉

C₂²×D₄

hypercenter

C₈×Q₈

D₅₉

hypocenter

C₅₉

#3358

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. #16838074	C₂	semidirect
2. Q₈×D₂₃₆	C₂	semidirect
3. #16837749	C₂	semidirect
4. #16837683	C₂	semidirect
5. #16837598	C₂	semidirect
6. C₄×D₄₇₂	C₂	semidirect
7. C₈×Q₈×C₅₉	C₂	semidirect
46. Q₈×C₅₉	D₈	semidirect
57. D₂₃₆	#41	semidirect
87. C₈×Q₈	D₅₉	semidirect
89. C₅₉	#3358	semidirect
129. Q₈	D₄₇₂	semidirect

H ≅ G/N

type

1. #16838074

C₂

semidirect

2. Q₈×D₂₃₆

C₂

semidirect

3. #16837749

C₂

semidirect

4. #16837683

C₂

semidirect

5. #16837598

C₂

semidirect

6. C₄×D₄₇₂

C₂

semidirect

7. C₈×Q₈×C₅₉

C₂

semidirect

46. Q₈×C₅₉

D₈

semidirect

57. D₂₃₆

#41

semidirect

87. C₈×Q₈

D₅₉

semidirect

89. C₅₉

#3358

semidirect

129. Q₈

D₄₇₂

semidirect

Subgroups — #21436732

Conjugacy classes of subgroups

Complements